Abstract:

The traditional curricula of Vector calculus at the Faculty of Sciences (Morelos State University, Mexico) has been revitalized by introducing Symbolic Computation System Maple in the teaching process for students of 4 semester studying Mathematics.
There has been noted the significant evolution of students’ representations aiming to construct the institutational representations.
A large set of graphic routines for visualizing complicated mathematical information, which complements the symbolical operations in Maple, helps to improve the students’ knowledge acquired during traditional education in order to transform them into consistent knowledge.
The use of computers facilitates the students‘ actions and allows them to operate information in a symbolical or algebraic manner, compute, display plots, animate results, and generate technical documents, easily accessed through the advanced worksheet.
In our talk we intent to restrict the presentation to the analysis of materialization– idealization processes (dialectic between ostensive and non-ostensive objects), applied to the problem-situations, which relates to variety of the properties of the “Boveda de Viviani”, that permit to reconstruct the fundamental concepts of curves, surfaces, formulas and algorithms to calculate longitudes of curves, areas of surfaces, volumes, etc, and clarify the concept of mapping.
These activities form preliminary knowledge to better understands the main topics of the course: where there is required to visualize the concepts such as vector fields, (its tangent and normal component, corresponding to the case of curve integral and the surface integral, respectively) vector operators, div, grad, rot, etc, which form the essential part the theory of vector calculus, the Stocks, Gauss, and Green’s theorem. In order to give a theoretical background and to enhance our teaching experimentation (as well as to convince the majority of academic staff, who reject the usage of calculators and computer facilities) we regard the variety of theoretical approaches. We detect that Maple is a magnificent tool for jointly analyzing the ostensive and non-ostensive objects intervenient in mathematical practice, showing the potential utility of the onto-semiotic approach (EOS Theory) to mathematical knowledge. The main argument in EOS- theory that the external representations
(Symbolic, graphical, linguistic, ostensive objects) are inevitable and dialectically accompanied by other non-ostensive mathematical objects and processes. As well, Maple facilitates the transformations and conversions of different systems of representations of mathematical objects (Duval 1995).This theory nowadays has the influence in education systems of some countries, for example, Hitt F (2004) considers that the semiotic representations in constructing and communicating mathematical knowledge are essential components of mathematical practices. The study and an analysis of the results obtained may lead to achieve the teaching guides which would offer effective methods, based on the visualization of symbols which represent mathematics objects, and manipulating different systems of semiotic representation.

Keywords:

semiotic representations, onto-semiotic approach, vector calculus.